This study presents a comprehensive spatial eigenanalysis of fully-discrete discontinuous spectral element methods, now generalizing previous spatial eigenanalysis that did not include time integration errors. The influence of discrete time integration is discussed in detail for different explicit Runge-Kutta (1st to 4th order accurate) schemes combined with either Discontinuous Galerkin (DG) or Spectral Difference (SD) methods, both here recovered from the Flux Reconstruction (FR) scheme. Selected numerical experiments using the improved SD method by Liang and Jameson [1] are performed to quantify the influence of time integration errors on actual simulations. These involve test cases of varied complexity, from one-dimensional linear advection equation studies to well-resolved and under-resolved inviscid vortical flows. It is shown that, while both well-resolved and under-resolved simulations of linear problems correlate well with the eigenanalysis prediction of time integration errors, the correlation can be much worse for under-resolved nonlinear problems. The effect of mesh regularity is also considered, where time integration errors are found to be, in the case of irregular grids, less pronounced than those of the spatial discretisation. In fact, for the under-resolved vortical flows considered, the predominance of spatial errors made it practically impossible for time integration errors to be distinctly identified. Nevertheless, for well-resolved nonlinear simulations, the effect of time integration errors could still be recognized. This highlights that the interaction between space and time discretisation errors is more complex than otherwise anticipated, contributing to the current understanding about when eigenanalysis can effectively predict the behaviour of numerical errors in practical under-resolved nonlinear problems, including under-resolved turbulence computations.

翻译：本研究对完全分解不连续的光谱元素错误进行了全面的空间天平分析,现在将先前的空间天平分析法(不包括时间整合误差)加以归纳,其中不包括时间整合错误。对于离散时间整合的影响,将详细讨论不同的显性龙格-库塔(第1至4级准确)计划的影响,结合不连续的伽列尔金(DG)或光谱差异(SD)方法,这两种方法均在此从通量重建(Flus Reform)方案(FR)中回收。使用Liang和Jameson[1]改进的SD方法进行的一些数字实验,以量化时间整合错误对实际模拟的影响。这些测试涉及从单维线性线性对映等方方程式研究到清晰解析和未彻底解析内流的复杂时间整合情况。在确定时间整合时间整合期间,对于时间整合时间整合的稳定性的不确定性可能比非线性非直线性问题要严重得多。在时间整合中,在确定时间整合过程中,对于时间整合的稳定性的稳定性也被视为不固定的不固定的数值。在时间整合中,在时间整合中发现,在时间整合中,在时间整合中,这些时间整合过程中的稳定性的稳定性的稳定性的稳定性的稳定性的稳定性的稳定性的稳定性的稳定性的稳定性的稳定性的稳定性的稳定性的稳定性的稳定性的稳定性的稳定性的稳定性的稳定性的稳定性的稳定性的稳定性的稳定性也比在时间流也会有助于。